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6 I ELEMENTS OF THE THEORY OF SETS

If, for every x e X, we have constructed an object T(x) which is an
element of Y, the relation y = T(x) is functional in ;;; the corresponding
mapping is written :c -> T(JC). This is of course the usual definition of a
mapping; it coincides essentially with the one given above, for if F is a
functional graph, it is the mapping x-*F(x). Examples (1.4.1) and (1.4.2)
are written respectively x-»b and x-» x. Other examples:

(1.4.3) The mapping Z -> X - Z of <P(X) into itself.

(1.4.4) The mappings z -* pr_x z of X x Y into X, and z -» pr₂ z of X x Y
into Y, which are called respectively the first and second projection in X x Y.

From the definition of equality of sets (Section 1.1) it follows that the
relation F = G between two mappings of X into Y is equivalent to the
relation " F(JC) = G(r) for every x e X."

If A is a subset of X, F a mapping of X into Y, the set F n (A x Y)
is a functional graph in A x Y, which, as a mapping, is called the restriction
ofF to A; when F and G have the same restriction to A (i.e. when F(x) = G(JC)
for every x e A) they are said to coincide in A. A mapping F of X into Y
having a given restriction F' to A is called an extension ofF' to X; there are
in general many different extensions of F'.

We will consider as an axiom (the "axiom of choice") the following
proposition:

(1.4.5) Given a mapping F of X into ^P(Y), such that F(x) ^ 0 for every
x e X, there exists a mapping fofX into Y such that f(x) e F(JC) for every
xeX.

It can sometimes be shown that a theorem proved with the help of the
axiom of choice can actually be proved without using that axiom. We shall
never go into such questions, which properly belong to a course in logic.

5. DIRECT AND INVERSE IMAGES

Let F be a mapping of X into Y. For any subset A of X, the subset
of Y defined by the property " there exists x e A such that y = F(x)" is
called the image (or direct image) of A by F and written F(A).

We have:

(15.1) F(A) = pr₂(Fn(AxY)).



	6 I ELEMENTS OF THE THEORY OF SETS If, for every x e X, we have constructed an object T(x) which is an element of Y, the relation y = T(x) is functional in ;;; the corresponding mapping is written :c -> T(JC). This is of course the usual definition of a mapping; it coincides essentially with the one given above, for if F is a functional graph, it is the mapping x-F(x).* Examples (1.4.1) and (1.4.2) are written respectively x-»b and x-» x. Other examples: (1.4.3) The mapping Z -> X - Z of <P(X) into itself. (1.4.4) The mappings z -* pr_x z of X x Y into X, and z -» pr₂ z of X x Y into Y, which are called respectively the first and second projection in X x Y. From the definition of equality of sets (Section 1.1) it follows that the relation F = G between two mappings of X into Y is equivalent to the relation " F(JC) = G(r) for every x e X." If A is a subset of X, F a mapping of X into Y, the set F n (A x Y) is a functional graph in A x Y, which, as a mapping, is called the restriction ofF to A; when F and G have the same restriction to A (i.e. when F(x) = G(JC) for every x e A) they are said to coincide in A. A mapping F of X into Y having a given restriction F' to A is called an extension ofF' to X; there are in general many different extensions of F'. We will consider as an axiom (the "axiom of choice") the following proposition:

	(1.4.5) Given a mapping F of X into ^P(Y), such that F(x) ^ 0 for every x e X, there exists a mapping fofX into Y such that f(x) e F(JC) for every xeX. It can sometimes be shown that a theorem proved with the help of the axiom of choice can actually be proved without using that axiom. We shall never go into such questions, which properly belong to a course in logic.

	5. DIRECT AND INVERSE IMAGES Let F be a mapping of X into Y. For any subset A of X, the subset of Y defined by the property " there exists x e A such that y = F(x)" is called the image (or direct image) of A by F and written F(A). We have: (15.1) F(A) = pr₂(Fn(AxY)).